1. Field of the Invention
The present invention relates generally to computer simulations using the finite element method, and more specifically, to a method for switching from an explicit Eulerian formulation to an implicit Lagrangian formulation in a finite element simulation of motion.
2. Related Art
The finite element method is a technique for obtaining approximate numerical solutions to boundary value problems which predict the response of physical systems subjected to external loads. The finite element method is described in detail by Thomas J. R. Hughes in “The Finite Element Method” (1987), published by Prentice-Hall, Inc., New Jersey, which is incorporated herein by this reference in its entirety. One common use of the finite element method is in the field of solid mechanics where it is used to analyze structural problems such as the formation of stamped sheet metal parts or the springback of stamped sheet metal parts. Another common application is injection molding. The equations describing the physical event of interest are generally overly complex to be solved exactly.
The finite element method is a technique where the geometry of the analysed structure is approximated as a set of points in space. The points, which are referred to as nodes, are connected together to form finite elements. This process is referred to as discretization. FIG. 1 shows the discretization of a wrench. The discretization of the geometry allows the differential equations to be approximated as finite sized matrix equations.
The finite element method can be used to run two or three dimensional simulations. In a two-dimensional (2D) simulation the elements are areas. In a three-dimensional (3D) simulation the elements are volumes. All of the simulations illustrated in this example are 3D simulations. The elements are therefore three dimensional volumes. However, for ease of illustration and explanation cross sections are used to illustrate the invention. The fill fraction, which in these 3D simulations is referred to as the volume fraction, would be an area fraction in the case of a 2D simulation. The elements and nodes form a mesh or grid, and these terms are used interchangeably throughout this application. Additionally, the elements are shown as cubes or rectangles, however other geometric shapes may be used.
In structural mechanics, the matrix equations describe the relationship between the stress and velocity fields and the acceleration field at a specific instant in time. To follow the deformation process, one needs to integrate the matrix equations in time. Due to non-linearities, an exact integration is generally not possible. A time discretization is necessary and one usually relies on a finite difference scheme to drive the solution forward in time. The matrix equations may be explicitly or implicitly integrated. Further explanation of explicit and implicit integration can be found in U.S. patent application Ser. No. 09/836,490 filed Apr. 17, 2001 entitled “Implicit-Explicit Switching for Finite Element Analysis”, which is hereby incorporated by reference in its entirety.
The finite element method has been implemented in finite element software, such as LS-DYNA, a commercial finite element analysis program developed by Livermore Software Technology Corporation of Livermore, Calif.
I. Explicit Eulerian Material and Void Formulation
An Eulerian finite element formulation is a method where the element grid is fixed in space. The material flows through the grid of elements. The material and void formulation refers to a special method where the elements are not necessarily filled with material. Each element contains a certain volume fraction(3D) or area fraction (2D) of material, ranging from 0 to 1.
Explicit time integration is suitable in situations where there are large non-linearities. No equation systems need to be solved and each time increment is computationally inexpensive. Small time increments are required in order to avoid numerical instabilities and diverging solutions. Explicit time integration techniques are well known in the art.
The sequence of pictures in FIG. 2 are snapshots from an Eulerian material and void simulation. It is a model of a high-speed impact where a piece of falling steel hits a rigid wall. The dark area indicates the location of the material. The steel plastically deforms upon impact. The first box shows the steel block just before impact, the middle box during impact, and the third box after impact has occurred. Observe that the element grid is fixed in space as the steel material flows.
An advantage of the Eulerian formulation is that there are no element distortions. Distorted elements are bad for the numerical accuracy and can even lead to crashing simulations.
A disadvantage is that the flux of material between elements (fixed in space) slows down the simulation and is bad for numerical accuracy. Another disadvantage is that the need for stability in the explicit time integration scheme puts a limit on the maximum allowed time step size. Therefore, slow and quasi-static processes, such as springback, are better treated with implicit methods.
II. Implicit Lagrangian Formulation
A Lagrangian Finite Element formulation is a method where the element grid follows the material flow. That is, each element represents the same piece of material throughout the complete simulation. FIG. 3 shows how the Lagrangian element grid follows the material flow.
Implicit time integration is suitable in static, quasi-static and in certain slow, dynamic simulations. An equation system needs to be solved for each time increment. Implicit time integration techniques are well known in the art.
An advantage of the Lagrangian formulation is that the numerical implementation is relatively straightforward. The numerical accuracy is good at small deformations. Another advantage is that certain implicit time integration schemes provide an unconditional stability. Therefore, large time steps can be allowed in quasi-static simulations.
A disadvantage of the Lagrangian formulation is that element distortions cause numerical problems when dealing with large deformations. Further, solving complex equation systems makes each time step rather costly in terms of time and computing resources.